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Some Quadratic Assignment Formulations and their impact on Gurobi - - PowerPoint PPT Presentation

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Agenda Problem Definition Branch and Bound Diagnosing and Improving our Formulation Conclusions Some Quadratic Assignment Formulations and their impact on Gurobi Leo Lopes 1 1 Monash University, School of Mathematical Sciences, Melbourne

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Agenda Problem Definition Branch and Bound Diagnosing and Improving our Formulation Conclusions

Some Quadratic Assignment Formulations and their impact on Gurobi

Leo Lopes1

1Monash University, School of Mathematical Sciences, Melbourne Australia

Monash University School of Mathematics

Lopes Monash University Some Quadratic Assignment Formulations and their impact on Gurobi 1/25

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Agenda Problem Definition Branch and Bound Diagnosing and Improving our Formulation Conclusions

Problem Definition Integer Programming Formulations The QAP A simple formulation Branch and Bound Diagnosing and Improving our Formulation Step 0: bound variables whenever possible Step 1: Strengthen cuts by adding more variables to each constraint Step 2: Smaller constants and more terms Conclusions

Lopes Monash University Some Quadratic Assignment Formulations and their impact on Gurobi 2/25

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Agenda Problem Definition Branch and Bound Diagnosing and Improving our Formulation Conclusions Integer Programming Formulations

The basic principle: strong and weak formulations

Lopes Monash University Some Quadratic Assignment Formulations and their impact on Gurobi 3/25

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Agenda Problem Definition Branch and Bound Diagnosing and Improving our Formulation Conclusions Integer Programming Formulations

The basic principle: formulations and branch-and-cut

Lopes Monash University Some Quadratic Assignment Formulations and their impact on Gurobi 4/25

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Agenda Problem Definition Branch and Bound Diagnosing and Improving our Formulation Conclusions The QAP

The Quadratic Assignment Problem (Input)

Distances: Flows: 1 2 3 4 5 1 2 2 4 2 4 4 1 3 3 2 4 3 5

Lopes Monash University Some Quadratic Assignment Formulations and their impact on Gurobi 5/25

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Agenda Problem Definition Branch and Bound Diagnosing and Improving our Formulation Conclusions The QAP

The Quadratic Assignment Problem (Output)

1 2 3 4 5 1 2 2 4 2 4 4 1 3 3 2 4 3 5

Lopes Monash University Some Quadratic Assignment Formulations and their impact on Gurobi 6/25

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Agenda Problem Definition Branch and Bound Diagnosing and Improving our Formulation Conclusions The QAP

Permutation Formulation of the QAP

Let:

◮ F be a set of facilities. Between each pair (f1,f2) of facilities, the flow

volume is vf1f2.

◮ L be a set of locations. The distance between location l1 and l2 is dl1l2.

Find an assignment g : F → L of each facility to a distinct location that minimizes the total distance in the network weighted by flow: min

g(x):F→L ∑ fi∈F ∑ f2∈F

vf1f2dg(f1),g(f2)

Lopes Monash University Some Quadratic Assignment Formulations and their impact on Gurobi 7/25

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Agenda Problem Definition Branch and Bound Diagnosing and Improving our Formulation Conclusions A simple formulation

Basic Mixed Integer Programming Formulation

Let:

◮ xfl =

  • 1

if facility f is assigned to location l

  • therwise

◮ yf1f2 ≥ 0 be the distance between facilities f1and f2. ◮ The objective: min x,y

∑

(f1,f2)∈F×F

vf1f2yf1f2

◮ Since the objective is lower when y is lower, a constraint that bounds the

distance between f1 and f2 from below to dl1l2 if and only if f1 is assigned to l1 and f2 is assigned to l2 will set y to the correct value:

◮ yf1f2 ≥ (xf1l1 + xf2l2 − 1)dl1l2,∀f1,f2 : f1 = f2,l1,l2 : l1 = l2

Lopes Monash University Some Quadratic Assignment Formulations and their impact on Gurobi 8/25

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Agenda Problem Definition Branch and Bound Diagnosing and Improving our Formulation Conclusions A simple formulation

In AMPL (1 of 2)

param n > 0; set FAC := 1..n; set LOC := 1..n; param d{LOC,LOC} >= 0; param v{FAC,FAC} >= 0; var Assign{FAC,LOC} binary; var Dist{FAC,FAC} >= 0;

Lopes Monash University Some Quadratic Assignment Formulations and their impact on Gurobi 9/25

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Agenda Problem Definition Branch and Bound Diagnosing and Improving our Formulation Conclusions A simple formulation

In AMPL (2 of 2)

minimize VolumeWeightedDistance: sumf1 in FAC,f2 in FAC: f2 <> f1 v[f1,f2]*Dist[f1,f2]; subject to Permute1f in FAC: suml in LOC Assign[f,l] = 1; subject to Permute2l in LOC: sumf in FAC Assign[f,l] = 1; subject to ComputeDistf1 in FAC, f2 in FAC,l1 in LOC, l2 in LOC: f1 <> f2 and l1 <> l2: Dist[f1,f2] >= (Assign[f1,l1] + Assign[f2,l2] - 1)*d[l1,l2];

Lopes Monash University Some Quadratic Assignment Formulations and their impact on Gurobi 10/25

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Agenda Problem Definition Branch and Bound Diagnosing and Improving our Formulation Conclusions A simple formulation

In AMPL (2 of 2)

minimize VolumeWeightedDistance: sumf1 in FAC,f2 in FAC: f2 <> f1 v[f1,f2]*Dist[f1,f2]; subject to Permute1f in FAC: suml in LOC Assign[f,l] = 1; subject to Permute2l in LOC: sumf in FAC Assign[f,l] = 1; subject to ComputeDistf1 in FAC, f2 in FAC,l1 in LOC, l2 in LOC: f1 <> f2 and l1 <> l2: Dist[f1,f2] >= (Assign[f1,l1] + Assign[f2,l2] - 1)*d[l1,l2];

Lopes Monash University Some Quadratic Assignment Formulations and their impact on Gurobi 10/25

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Agenda Problem Definition Branch and Bound Diagnosing and Improving our Formulation Conclusions A simple formulation

In AMPL (2 of 2)

minimize VolumeWeightedDistance: sumf1 in FAC,f2 in FAC: f2 <> f1 v[f1,f2]*Dist[f1,f2]; subject to Permute1f in FAC: suml in LOC Assign[f,l] = 1; subject to Permute2l in LOC: sumf in FAC Assign[f,l] = 1; subject to ComputeDistf1 in FAC, f2 in FAC,l1 in LOC, l2 in LOC: f1 <> f2 and l1 <> l2: Dist[f1,f2] >= (Assign[f1,l1] + Assign[f2,l2] - 1)*d[l1,l2];

Lopes Monash University Some Quadratic Assignment Formulations and their impact on Gurobi 10/25

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Agenda Problem Definition Branch and Bound Diagnosing and Improving our Formulation Conclusions A simple formulation

Solver Results The math is correct, but it “doesn’t work” “in practice”!

  • ption solver gurobix;
  • ption gurobi_options $gurobi_options ’outlev 1’;

reset; model qapNaive.ampl; read n, {f in FAC, l in LOC} d[f,l], {f in FAC, l in LOC} v[f,l] < instances/nug12.dat;

The branch-and-bound tree will grow until it consumes all memory available (2010).

Producing good bounds is essential

Let’s see the details.

Lopes Monash University Some Quadratic Assignment Formulations and their impact on Gurobi 11/25

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Agenda Problem Definition Branch and Bound Diagnosing and Improving our Formulation Conclusions

A Branch and Bound Tree

After node Lower Bound Upper Bound 2

∞

1 2

∞

2 3 4

∞

4 5

∞

5 5

∞

6

∞

7* 8 6 8 9 6 7 10*

Lopes Monash University Some Quadratic Assignment Formulations and their impact on Gurobi 12/25

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Agenda Problem Definition Branch and Bound Diagnosing and Improving our Formulation Conclusions

Reading the Gurobi Log

Gurobi 3.0.0: outlev 1 Optimize a model with 17448 Rows, 276 Columns and 52560 NonZeros Presolve removed 5544 rows and 42 columns Presolve time: 0.57s Presolved: 11904 Rows, 234 Columns, 47808 Nonzeros Objective GCD is 1 Found heuristic solution: objective 868.0000000 Found heuristic solution: objective 724.0000000 Root relaxation: objective 0.000000e+00, 768 iterations, 0.05 seconds Nodes | Current Node | Objective Bounds | Work Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time ... 311942 84869 cutoff 42 582.00000 499.14837 14.2% 20.9 230s 321368 85757 575.74359 48 46 582.00000 500.51202 14.0% 20.8 235s 331149 86576 523.66648 43 65 582.00000 501.93019 13.8% 20.7 240s *332324 85008 39 578.0000000 502.07250 13.1% 20.6 240s ...

Lopes Monash University Some Quadratic Assignment Formulations and their impact on Gurobi 13/25

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Agenda Problem Definition Branch and Bound Diagnosing and Improving our Formulation Conclusions

Reading the Gurobi Log

Gurobi 3.0.0: outlev 1 Optimize a model with 17448 Rows, 276 Columns and 52560 NonZeros Presolve removed 5544 rows and 42 columns Presolve time: 0.57s Presolved: 11904 Rows, 234 Columns, 47808 Nonzeros Objective GCD is 1 Found heuristic solution: objective 868.0000000 Found heuristic solution: objective 724.0000000 Root relaxation: objective 0.000000e+00, 768 iterations, 0.05 seconds Nodes | Current Node | Objective Bounds | Work Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time ... 311942 84869 cutoff 42 582.00000 499.14837 14.2% 20.9 230s 321368 85757 575.74359 48 46 582.00000 500.51202 14.0% 20.8 235s 331149 86576 523.66648 43 65 582.00000 501.93019 13.8% 20.7 240s *332324 85008 39 578.0000000 502.07250 13.1% 20.6 240s ...

Lopes Monash University Some Quadratic Assignment Formulations and their impact on Gurobi 13/25

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Agenda Problem Definition Branch and Bound Diagnosing and Improving our Formulation Conclusions

Reading the Gurobi Log

Gurobi 3.0.0: outlev 1 Optimize a model with 17448 Rows, 276 Columns and 52560 NonZeros Presolve removed 5544 rows and 42 columns Presolve time: 0.57s Presolved: 11904 Rows, 234 Columns, 47808 Nonzeros Objective GCD is 1 Found heuristic solution: objective 868.0000000 Found heuristic solution: objective 724.0000000 Root relaxation: objective 0.000000e+00, 768 iterations, 0.05 seconds Nodes | Current Node | Objective Bounds | Work Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time ... 311942 84869 cutoff 42 582.00000 499.14837 14.2% 20.9 230s 321368 85757 575.74359 48 46 582.00000 500.51202 14.0% 20.8 235s 331149 86576 523.66648 43 65 582.00000 501.93019 13.8% 20.7 240s *332324 85008 39 578.0000000 502.07250 13.1% 20.6 240s ...

Lopes Monash University Some Quadratic Assignment Formulations and their impact on Gurobi 13/25

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Agenda Problem Definition Branch and Bound Diagnosing and Improving our Formulation Conclusions

Reading the Gurobi Log

Gurobi 3.0.0: outlev 1 Optimize a model with 17448 Rows, 276 Columns and 52560 NonZeros Presolve removed 5544 rows and 42 columns Presolve time: 0.57s Presolved: 11904 Rows, 234 Columns, 47808 Nonzeros Objective GCD is 1 Found heuristic solution: objective 868.0000000 Found heuristic solution: objective 724.0000000 Root relaxation: objective 0.000000e+00, 768 iterations, 0.05 seconds Nodes | Current Node | Objective Bounds | Work Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time ... 311942 84869 cutoff 42 582.00000 499.14837 14.2% 20.9 230s 321368 85757 575.74359 48 46 582.00000 500.51202 14.0% 20.8 235s 331149 86576 523.66648 43 65 582.00000 501.93019 13.8% 20.7 240s *332324 85008 39 578.0000000 502.07250 13.1% 20.6 240s ...

Lopes Monash University Some Quadratic Assignment Formulations and their impact on Gurobi 13/25

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Agenda Problem Definition Branch and Bound Diagnosing and Improving our Formulation Conclusions

Reading the Gurobi Log

Gurobi 3.0.0: outlev 1 Optimize a model with 17448 Rows, 276 Columns and 52560 NonZeros Presolve removed 5544 rows and 42 columns Presolve time: 0.57s Presolved: 11904 Rows, 234 Columns, 47808 Nonzeros Objective GCD is 1 Found heuristic solution: objective 868.0000000 Found heuristic solution: objective 724.0000000 Root relaxation: objective 0.000000e+00, 768 iterations, 0.05 seconds Nodes | Current Node | Objective Bounds | Work Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time ... 311942 84869 cutoff 42 582.00000 499.14837 14.2% 20.9 230s 321368 85757 575.74359 48 46 582.00000 500.51202 14.0% 20.8 235s 331149 86576 523.66648 43 65 582.00000 501.93019 13.8% 20.7 240s *332324 85008 39 578.0000000 502.07250 13.1% 20.6 240s ...

Lopes Monash University Some Quadratic Assignment Formulations and their impact on Gurobi 13/25

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Agenda Problem Definition Branch and Bound Diagnosing and Improving our Formulation Conclusions

Reading the Gurobi Log

Gurobi 3.0.0: outlev 1 Optimize a model with 17448 Rows, 276 Columns and 52560 NonZeros Presolve removed 5544 rows and 42 columns Presolve time: 0.57s Presolved: 11904 Rows, 234 Columns, 47808 Nonzeros Objective GCD is 1 Found heuristic solution: objective 868.0000000 Found heuristic solution: objective 724.0000000 Root relaxation: objective 0.000000e+00, 768 iterations, 0.05 seconds Nodes | Current Node | Objective Bounds | Work Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time ... 311942 84869 cutoff 42 582.00000 499.14837 14.2% 20.9 230s 321368 85757 575.74359 48 46 582.00000 500.51202 14.0% 20.8 235s 331149 86576 523.66648 43 65 582.00000 501.93019 13.8% 20.7 240s *332324 85008 39 578.0000000 502.07250 13.1% 20.6 240s ...

Lopes Monash University Some Quadratic Assignment Formulations and their impact on Gurobi 13/25

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Agenda Problem Definition Branch and Bound Diagnosing and Improving our Formulation Conclusions

Reading the Gurobi Log

Gurobi 3.0.0: outlev 1 Optimize a model with 17448 Rows, 276 Columns and 52560 NonZeros Presolve removed 5544 rows and 42 columns Presolve time: 0.57s Presolved: 11904 Rows, 234 Columns, 47808 Nonzeros Objective GCD is 1 Found heuristic solution: objective 868.0000000 Found heuristic solution: objective 724.0000000 Root relaxation: objective 0.000000e+00, 768 iterations, 0.05 seconds Nodes | Current Node | Objective Bounds | Work Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time ... 311942 84869 cutoff 42 582.00000 499.14837 14.2% 20.9 230s 321368 85757 575.74359 48 46 582.00000 500.51202 14.0% 20.8 235s 331149 86576 523.66648 43 65 582.00000 501.93019 13.8% 20.7 240s *332324 85008 39 578.0000000 502.07250 13.1% 20.6 240s ...

Lopes Monash University Some Quadratic Assignment Formulations and their impact on Gurobi 13/25

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Agenda Problem Definition Branch and Bound Diagnosing and Improving our Formulation Conclusions

Reading the Gurobi Log

Gurobi 3.0.0: outlev 1 Optimize a model with 17448 Rows, 276 Columns and 52560 NonZeros Presolve removed 5544 rows and 42 columns Presolve time: 0.57s Presolved: 11904 Rows, 234 Columns, 47808 Nonzeros Objective GCD is 1 Found heuristic solution: objective 868.0000000 Found heuristic solution: objective 724.0000000 Root relaxation: objective 0.000000e+00, 768 iterations, 0.05 seconds Nodes | Current Node | Objective Bounds | Work Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time ... 311942 84869 cutoff 42 582.00000 499.14837 14.2% 20.9 230s 321368 85757 575.74359 48 46 582.00000 500.51202 14.0% 20.8 235s 331149 86576 523.66648 43 65 582.00000 501.93019 13.8% 20.7 240s *332324 85008 39 578.0000000 502.07250 13.1% 20.6 240s ...

Lopes Monash University Some Quadratic Assignment Formulations and their impact on Gurobi 13/25

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Agenda Problem Definition Branch and Bound Diagnosing and Improving our Formulation Conclusions

Reading the Gurobi Log

Gurobi 3.0.0: outlev 1 Optimize a model with 17448 Rows, 276 Columns and 52560 NonZeros Presolve removed 5544 rows and 42 columns Presolve time: 0.57s Presolved: 11904 Rows, 234 Columns, 47808 Nonzeros Objective GCD is 1 Found heuristic solution: objective 868.0000000 Found heuristic solution: objective 724.0000000 Root relaxation: objective 0.000000e+00, 768 iterations, 0.05 seconds Nodes | Current Node | Objective Bounds | Work Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time ... 311942 84869 cutoff 42 582.00000 499.14837 14.2% 20.9 230s 321368 85757 575.74359 48 46 582.00000 500.51202 14.0% 20.8 235s 331149 86576 523.66648 43 65 582.00000 501.93019 13.8% 20.7 240s *332324 85008 39 578.0000000 502.07250 13.1% 20.6 240s ...

Lopes Monash University Some Quadratic Assignment Formulations and their impact on Gurobi 13/25

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Agenda Problem Definition Branch and Bound Diagnosing and Improving our Formulation Conclusions

Diagnosing weak bounds

... Nodes | Current Node | Objective Bounds | Work Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time 0.00000 37 724.00000 0.00000 100%

  • 1s

0.00000 40 724.00000 0.00000 100%

  • 1s

... 0.00000 31 724.00000 0.00000 100%

  • 4s

H 682.0000000 0.00000 100%

  • 4s

0.00000 31 682.00000 0.00000 100%

  • 4s

... H 1238 974 618.0000000 15.97409 97.4% 53.8 32s 1320 1020 142.41196 15 107 618.00000 27.18269 95.6% 53.1 35s H 1369 1048 606.0000000 27.18269 95.5% 52.7 35s 1470 1131 377.68530 33 40 606.00000 27.75623 95.4% 52.3 41s 1473 1135 27.75623 16 49 606.00000 27.75623 95.4% 53.6 47s 1474 1135 27.75623 16 96 606.00000 27.75623 95.4% 53.7 50s ...

Lopes Monash University Some Quadratic Assignment Formulations and their impact on Gurobi 14/25

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Agenda Problem Definition Branch and Bound Diagnosing and Improving our Formulation Conclusions

Diagnosing weak bounds

... Nodes | Current Node | Objective Bounds | Work Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time 0.00000 37 724.00000 0.00000 100%

  • 1s

0.00000 40 724.00000 0.00000 100%

  • 1s

... 0.00000 31 724.00000 0.00000 100%

  • 4s

H 682.0000000 0.00000 100%

  • 4s

0.00000 31 682.00000 0.00000 100%

  • 4s

... H 1238 974 618.0000000 15.97409 97.4% 53.8 32s 1320 1020 142.41196 15 107 618.00000 27.18269 95.6% 53.1 35s H 1369 1048 606.0000000 27.18269 95.5% 52.7 35s 1470 1131 377.68530 33 40 606.00000 27.75623 95.4% 52.3 41s 1473 1135 27.75623 16 49 606.00000 27.75623 95.4% 53.6 47s 1474 1135 27.75623 16 96 606.00000 27.75623 95.4% 53.7 50s ...

Lopes Monash University Some Quadratic Assignment Formulations and their impact on Gurobi 14/25

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Agenda Problem Definition Branch and Bound Diagnosing and Improving our Formulation Conclusions Step 0: bound variables whenever possible

Observation on variable bounds

No matter what facility gets assigned where, the distance between any two facilities must be at least the minimal distance between any two locations:

◮ yf1f2 ≥ min(l1,l2)∈L×L:l1=l2 dl1l2 ◮ In AMPL:

var Dist{FAC,FAC} >= min{l1 in LOC, l2 in LOC: l1 <> l2} d[l1,l2];

Lopes Monash University Some Quadratic Assignment Formulations and their impact on Gurobi 15/25

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Agenda Problem Definition Branch and Bound Diagnosing and Improving our Formulation Conclusions Step 0: bound variables whenever possible

Solver Results At least it solves, but...

Explored 10,180,956 nodes (160,414,000 simplex iterations) in 13,664.80 seconds Thread count was 4 (of 4 available processors)

Let’s see the details.

Lopes Monash University Some Quadratic Assignment Formulations and their impact on Gurobi 16/25

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Agenda Problem Definition Branch and Bound Diagnosing and Improving our Formulation Conclusions Step 0: bound variables whenever possible

Initial bounds vs. the muddled middle

... Nodes | Current Node | Objective Bounds | Work Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time ... 348.00000 27 724.00000 348.00000 51.9%

  • 0s

... H 5117 3068 622.0000000 353.58269 43.2% 28.2 61s * 6157 3790 38 612.0000000 354.34464 42.1% 27.6 64s 6179 3811 382.04491 29 78 612.00000 354.34464 42.1% 27.6 65s ... 146583 88509 447.72035 31 66 590.00000 402.72441 31.7% 23.6 600s 147934 89260 549.35810 41 65 590.00000 402.91520 31.7% 23.6 605s 149758 90263 516.43027 44 67 590.00000 403.18118 31.7% 23.6 610s ... 1660356 685638 563.64870 59 55 586.00000 469.62771 19.9% 19.7 3000s 1664829 687024 cutoff 48 586.00000 469.71047 19.8% 19.7 3005s ...

Lopes Monash University Some Quadratic Assignment Formulations and their impact on Gurobi 17/25

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Agenda Problem Definition Branch and Bound Diagnosing and Improving our Formulation Conclusions Step 0: bound variables whenever possible

Initial bounds vs. the muddled middle

... Nodes | Current Node | Objective Bounds | Work Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time ... 348.00000 27 724.00000 348.00000 51.9%

  • 0s

... H 5117 3068 622.0000000 353.58269 43.2% 28.2 61s * 6157 3790 38 612.0000000 354.34464 42.1% 27.6 64s 6179 3811 382.04491 29 78 612.00000 354.34464 42.1% 27.6 65s ... 146583 88509 447.72035 31 66 590.00000 402.72441 31.7% 23.6 600s 147934 89260 549.35810 41 65 590.00000 402.91520 31.7% 23.6 605s 149758 90263 516.43027 44 67 590.00000 403.18118 31.7% 23.6 610s ... 1660356 685638 563.64870 59 55 586.00000 469.62771 19.9% 19.7 3000s 1664829 687024 cutoff 48 586.00000 469.71047 19.8% 19.7 3005s ...

Lopes Monash University Some Quadratic Assignment Formulations and their impact on Gurobi 17/25

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Agenda Problem Definition Branch and Bound Diagnosing and Improving our Formulation Conclusions Step 0: bound variables whenever possible

Initial bounds vs. the muddled middle

... Nodes | Current Node | Objective Bounds | Work Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time ... 348.00000 27 724.00000 348.00000 51.9%

  • 0s

... H 5117 3068 622.0000000 353.58269 43.2% 28.2 61s * 6157 3790 38 612.0000000 354.34464 42.1% 27.6 64s 6179 3811 382.04491 29 78 612.00000 354.34464 42.1% 27.6 65s ... 146583 88509 447.72035 31 66 590.00000 402.72441 31.7% 23.6 600s 147934 89260 549.35810 41 65 590.00000 402.91520 31.7% 23.6 605s 149758 90263 516.43027 44 67 590.00000 403.18118 31.7% 23.6 610s ... 1660356 685638 563.64870 59 55 586.00000 469.62771 19.9% 19.7 3000s 1664829 687024 cutoff 48 586.00000 469.71047 19.8% 19.7 3005s ...

Lopes Monash University Some Quadratic Assignment Formulations and their impact on Gurobi 17/25

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Agenda Problem Definition Branch and Bound Diagnosing and Improving our Formulation Conclusions Step 0: bound variables whenever possible

Initial bounds vs. the muddled middle

... Nodes | Current Node | Objective Bounds | Work Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time ... 348.00000 27 724.00000 348.00000 51.9%

  • 0s

... H 5117 3068 622.0000000 353.58269 43.2% 28.2 61s * 6157 3790 38 612.0000000 354.34464 42.1% 27.6 64s 6179 3811 382.04491 29 78 612.00000 354.34464 42.1% 27.6 65s ... 146583 88509 447.72035 31 66 590.00000 402.72441 31.7% 23.6 600s 147934 89260 549.35810 41 65 590.00000 402.91520 31.7% 23.6 605s 149758 90263 516.43027 44 67 590.00000 403.18118 31.7% 23.6 610s ... 1660356 685638 563.64870 59 55 586.00000 469.62771 19.9% 19.7 3000s 1664829 687024 cutoff 48 586.00000 469.71047 19.8% 19.7 3005s ...

Lopes Monash University Some Quadratic Assignment Formulations and their impact on Gurobi 17/25

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Agenda Problem Definition Branch and Bound Diagnosing and Improving our Formulation Conclusions Step 0: bound variables whenever possible

Initial bounds vs. the muddled middle

... Nodes | Current Node | Objective Bounds | Work Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time ... 348.00000 27 724.00000 348.00000 51.9%

  • 0s

... H 5117 3068 622.0000000 353.58269 43.2% 28.2 61s * 6157 3790 38 612.0000000 354.34464 42.1% 27.6 64s 6179 3811 382.04491 29 78 612.00000 354.34464 42.1% 27.6 65s ... 146583 88509 447.72035 31 66 590.00000 402.72441 31.7% 23.6 600s 147934 89260 549.35810 41 65 590.00000 402.91520 31.7% 23.6 605s 149758 90263 516.43027 44 67 590.00000 403.18118 31.7% 23.6 610s ... 1660356 685638 563.64870 59 55 586.00000 469.62771 19.9% 19.7 3000s 1664829 687024 cutoff 48 586.00000 469.71047 19.8% 19.7 3005s ...

Lopes Monash University Some Quadratic Assignment Formulations and their impact on Gurobi 17/25

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Agenda Problem Definition Branch and Bound Diagnosing and Improving our Formulation Conclusions Step 1: Strengthen cuts by adding more variables to each constraint

Rule of thumb: use more variables in each constraint

If we use more variables in the constraints, when some of them get fixed, the resulting LP bounds tend to be tigher.

Observation:

If we assume that f1 was assigned to l1, then for each f2: the distance between f1 and f2 must be at least the sum of the distances out of l1 minus the sum of all the other distances except that between l1 and l2 when f2 is assigned to l2:

◮ yf1f2 ≥ ∑l2=l1 dl1l2xf1l1 −∑f2=f1(∑l3∈{l1,l2} dl1l3)xf2l2,∀f1,f2,l1 : f1 = f2 ◮ In AMPL:

subject to ComputeDist2{f1 in FAC, f2 in FAC,l1 in LOC: f1 <> f2}: Dist[f1,f2] >= Assign[f1,l1]*sum{l2 in LOC: l2 <> l1} d[l1,l2]

  • sum{l2 in LOC: l2 <> l1}(Assign[f2,l2] *

sum{l3 in LOC: l3 <> l1 and l3 <> l2} d[l1,l3] ) ;

Lopes Monash University Some Quadratic Assignment Formulations and their impact on Gurobi 18/25

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SLIDE 34

Agenda Problem Definition Branch and Bound Diagnosing and Improving our Formulation Conclusions Step 1: Strengthen cuts by adding more variables to each constraint

Solver Results Much better...

Explored 2,855,769 nodes (19,376,754 simplex iterations) in 800.96 seconds Thread count was 4 (of 4 available processors)

Much better. Let’s see the details.

Lopes Monash University Some Quadratic Assignment Formulations and their impact on Gurobi 19/25

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SLIDE 35

Agenda Problem Definition Branch and Bound Diagnosing and Improving our Formulation Conclusions Step 1: Strengthen cuts by adding more variables to each constraint

A healthy tree

... Nodes | Current Node | Objective Bounds | Work Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time ... 148565 58745 472.38928 42 41 578.00000 426.17068 26.3% 11.0 60s 162715 63069 550.49347 51 41 578.00000 428.39964 25.9% 10.9 65s 176814 67294 493.66237 43 41 578.00000 430.60198 25.5% 10.8 70s ... 2352964 205490 cutoff 32 578.00000 525.24585 9.13% 7.3 600s 2374875 201750 cutoff 47 578.00000 526.14261 8.97% 7.3 605s 2398390 197345 536.17206 41 43 578.00000 527.17244 8.79% 7.3 610s ...

Lopes Monash University Some Quadratic Assignment Formulations and their impact on Gurobi 20/25

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SLIDE 36

Agenda Problem Definition Branch and Bound Diagnosing and Improving our Formulation Conclusions Step 1: Strengthen cuts by adding more variables to each constraint

A healthy tree

... Nodes | Current Node | Objective Bounds | Work Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time ... 148565 58745 472.38928 42 41 578.00000 426.17068 26.3% 11.0 60s 162715 63069 550.49347 51 41 578.00000 428.39964 25.9% 10.9 65s 176814 67294 493.66237 43 41 578.00000 430.60198 25.5% 10.8 70s ... 2352964 205490 cutoff 32 578.00000 525.24585 9.13% 7.3 600s 2374875 201750 cutoff 47 578.00000 526.14261 8.97% 7.3 605s 2398390 197345 536.17206 41 43 578.00000 527.17244 8.79% 7.3 610s ...

Lopes Monash University Some Quadratic Assignment Formulations and their impact on Gurobi 20/25

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SLIDE 37

Agenda Problem Definition Branch and Bound Diagnosing and Improving our Formulation Conclusions Step 1: Strengthen cuts by adding more variables to each constraint

A healthy tree

... Nodes | Current Node | Objective Bounds | Work Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time ... 148565 58745 472.38928 42 41 578.00000 426.17068 26.3% 11.0 60s 162715 63069 550.49347 51 41 578.00000 428.39964 25.9% 10.9 65s 176814 67294 493.66237 43 41 578.00000 430.60198 25.5% 10.8 70s ... 2352964 205490 cutoff 32 578.00000 525.24585 9.13% 7.3 600s 2374875 201750 cutoff 47 578.00000 526.14261 8.97% 7.3 605s 2398390 197345 536.17206 41 43 578.00000 527.17244 8.79% 7.3 610s ...

Lopes Monash University Some Quadratic Assignment Formulations and their impact on Gurobi 20/25

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SLIDE 38

Agenda Problem Definition Branch and Bound Diagnosing and Improving our Formulation Conclusions Step 1: Strengthen cuts by adding more variables to each constraint

A healthy tree

... Nodes | Current Node | Objective Bounds | Work Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time ... 148565 58745 472.38928 42 41 578.00000 426.17068 26.3% 11.0 60s 162715 63069 550.49347 51 41 578.00000 428.39964 25.9% 10.9 65s 176814 67294 493.66237 43 41 578.00000 430.60198 25.5% 10.8 70s ... 2352964 205490 cutoff 32 578.00000 525.24585 9.13% 7.3 600s 2374875 201750 cutoff 47 578.00000 526.14261 8.97% 7.3 605s 2398390 197345 536.17206 41 43 578.00000 527.17244 8.79% 7.3 610s ...

Lopes Monash University Some Quadratic Assignment Formulations and their impact on Gurobi 20/25

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SLIDE 39

Agenda Problem Definition Branch and Bound Diagnosing and Improving our Formulation Conclusions Step 1: Strengthen cuts by adding more variables to each constraint

A healthy tree

... Nodes | Current Node | Objective Bounds | Work Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time ... 148565 58745 472.38928 42 41 578.00000 426.17068 26.3% 11.0 60s 162715 63069 550.49347 51 41 578.00000 428.39964 25.9% 10.9 65s 176814 67294 493.66237 43 41 578.00000 430.60198 25.5% 10.8 70s ... 2352964 205490 cutoff 32 578.00000 525.24585 9.13% 7.3 600s 2374875 201750 cutoff 47 578.00000 526.14261 8.97% 7.3 605s 2398390 197345 536.17206 41 43 578.00000 527.17244 8.79% 7.3 610s ...

Lopes Monash University Some Quadratic Assignment Formulations and their impact on Gurobi 20/25

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SLIDE 40

Agenda Problem Definition Branch and Bound Diagnosing and Improving our Formulation Conclusions Step 2: Smaller constants and more terms

Smaller constants tend to be better

◮ The sums over all distances are likely making many of the constraints

non-binding in the LP relaxations. A formulation with smaller numbers will usually perform better.

◮ Since ∑l xfl = 1, then either xfl = 1 or ∑l2=l xfl2 = 1. In the former case,

yf1f2 will be dll2. In the latter case, then discounting maxl2=l dll2 from yf1f2 is just enough to make the constraint non-binding.

◮ In algebra:

yf1f2 ≥ ∑l2=l1 dl1l2xf2l2 −(maxl2=l1 dl1l2)∑l2=l1 xf1l2,∀f1,f2,l1 : f1 = f2

◮ Rearranging and reorganizing terms: Ml = maxl2=l dll2 and

yf1f2 ≥ ∑l2=l(dll2xf2l2 − Mlxf1l2),∀f1,f2,l : f1 = f2

◮ In AMPL:

param maxd{l in LOC} := max{l2 in LOC: l2 <> l} d[l,l2]; subject to ComputeDist3{f1 in FAC, f2 in FAC,l in LOC: f1 <> f2}: Dist[f1,f2] >= sum{l2 in LOC: l2 <> l} ( d[l,l2]*Assign[f2,l2] - maxd[l]* Assign[f1,l2] );

Lopes Monash University Some Quadratic Assignment Formulations and their impact on Gurobi 21/25

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SLIDE 41

Agenda Problem Definition Branch and Bound Diagnosing and Improving our Formulation Conclusions Step 2: Smaller constants and more terms

Solver Results Total time cut in half again...

Explored 723,627 nodes (12,293,156 simplex iterations) in 428.08 seconds Thread count was 4 (of 4 available processors)

Much better again. Let’s see the details.

Lopes Monash University Some Quadratic Assignment Formulations and their impact on Gurobi 22/25

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SLIDE 42

Agenda Problem Definition Branch and Bound Diagnosing and Improving our Formulation Conclusions Step 2: Smaller constants and more terms

A healthier tree

... Nodes | Current Node | Objective Bounds | Work Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time ... 41971 20620 578.17341 42 57 586.00000 426.84741 27.2% 26.7 60s 49310 23712 506.64911 34 68 586.00000 430.28747 26.6% 26.2 65s 56491 26513 580.49821 55 64 586.00000 433.86992 26.0% 25.9 70s ... 443823 85925 cutoff 35 578.00000 517.55057 10.5% 19.6 300s 454929 85247 572.52381 38 46 578.00000 518.97396 10.2% 19.5 305s 461098 84930 cutoff 29 578.00000 519.80133 10.1% 19.4 310s ... 676266 34174 cutoff 39 578.00000 552.77727 4.36% 17.6 410s 687701 27147 cutoff 50 578.00000 556.03894 3.80% 17.5 415s 699749 18846 cutoff 30 578.00000 560.31293 3.06% 17.4 420s 713063 8885 568.11389 37 59 578.00000 566.75240 1.95% 17.2 425s Explored 723627 nodes (12293156 simplex iterations) in 428.08 seconds Thread count was 4 (of 4 available processors)

Lopes Monash University Some Quadratic Assignment Formulations and their impact on Gurobi 23/25

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SLIDE 43

Agenda Problem Definition Branch and Bound Diagnosing and Improving our Formulation Conclusions Step 2: Smaller constants and more terms

A healthier tree

... Nodes | Current Node | Objective Bounds | Work Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time ... 41971 20620 578.17341 42 57 586.00000 426.84741 27.2% 26.7 60s 49310 23712 506.64911 34 68 586.00000 430.28747 26.6% 26.2 65s 56491 26513 580.49821 55 64 586.00000 433.86992 26.0% 25.9 70s ... 443823 85925 cutoff 35 578.00000 517.55057 10.5% 19.6 300s 454929 85247 572.52381 38 46 578.00000 518.97396 10.2% 19.5 305s 461098 84930 cutoff 29 578.00000 519.80133 10.1% 19.4 310s ... 676266 34174 cutoff 39 578.00000 552.77727 4.36% 17.6 410s 687701 27147 cutoff 50 578.00000 556.03894 3.80% 17.5 415s 699749 18846 cutoff 30 578.00000 560.31293 3.06% 17.4 420s 713063 8885 568.11389 37 59 578.00000 566.75240 1.95% 17.2 425s Explored 723627 nodes (12293156 simplex iterations) in 428.08 seconds Thread count was 4 (of 4 available processors)

Lopes Monash University Some Quadratic Assignment Formulations and their impact on Gurobi 23/25

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SLIDE 44

Agenda Problem Definition Branch and Bound Diagnosing and Improving our Formulation Conclusions Step 2: Smaller constants and more terms

A healthier tree

... Nodes | Current Node | Objective Bounds | Work Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time ... 41971 20620 578.17341 42 57 586.00000 426.84741 27.2% 26.7 60s 49310 23712 506.64911 34 68 586.00000 430.28747 26.6% 26.2 65s 56491 26513 580.49821 55 64 586.00000 433.86992 26.0% 25.9 70s ... 443823 85925 cutoff 35 578.00000 517.55057 10.5% 19.6 300s 454929 85247 572.52381 38 46 578.00000 518.97396 10.2% 19.5 305s 461098 84930 cutoff 29 578.00000 519.80133 10.1% 19.4 310s ... 676266 34174 cutoff 39 578.00000 552.77727 4.36% 17.6 410s 687701 27147 cutoff 50 578.00000 556.03894 3.80% 17.5 415s 699749 18846 cutoff 30 578.00000 560.31293 3.06% 17.4 420s 713063 8885 568.11389 37 59 578.00000 566.75240 1.95% 17.2 425s Explored 723627 nodes (12293156 simplex iterations) in 428.08 seconds Thread count was 4 (of 4 available processors)

Lopes Monash University Some Quadratic Assignment Formulations and their impact on Gurobi 23/25

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SLIDE 45

Agenda Problem Definition Branch and Bound Diagnosing and Improving our Formulation Conclusions Step 2: Smaller constants and more terms

A healthier tree

... Nodes | Current Node | Objective Bounds | Work Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time ... 41971 20620 578.17341 42 57 586.00000 426.84741 27.2% 26.7 60s 49310 23712 506.64911 34 68 586.00000 430.28747 26.6% 26.2 65s 56491 26513 580.49821 55 64 586.00000 433.86992 26.0% 25.9 70s ... 443823 85925 cutoff 35 578.00000 517.55057 10.5% 19.6 300s 454929 85247 572.52381 38 46 578.00000 518.97396 10.2% 19.5 305s 461098 84930 cutoff 29 578.00000 519.80133 10.1% 19.4 310s ... 676266 34174 cutoff 39 578.00000 552.77727 4.36% 17.6 410s 687701 27147 cutoff 50 578.00000 556.03894 3.80% 17.5 415s 699749 18846 cutoff 30 578.00000 560.31293 3.06% 17.4 420s 713063 8885 568.11389 37 59 578.00000 566.75240 1.95% 17.2 425s Explored 723627 nodes (12293156 simplex iterations) in 428.08 seconds Thread count was 4 (of 4 available processors)

Lopes Monash University Some Quadratic Assignment Formulations and their impact on Gurobi 23/25

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SLIDE 46

Agenda Problem Definition Branch and Bound Diagnosing and Improving our Formulation Conclusions Step 2: Smaller constants and more terms

A healthier tree

... Nodes | Current Node | Objective Bounds | Work Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time ... 41971 20620 578.17341 42 57 586.00000 426.84741 27.2% 26.7 60s 49310 23712 506.64911 34 68 586.00000 430.28747 26.6% 26.2 65s 56491 26513 580.49821 55 64 586.00000 433.86992 26.0% 25.9 70s ... 443823 85925 cutoff 35 578.00000 517.55057 10.5% 19.6 300s 454929 85247 572.52381 38 46 578.00000 518.97396 10.2% 19.5 305s 461098 84930 cutoff 29 578.00000 519.80133 10.1% 19.4 310s ... 676266 34174 cutoff 39 578.00000 552.77727 4.36% 17.6 410s 687701 27147 cutoff 50 578.00000 556.03894 3.80% 17.5 415s 699749 18846 cutoff 30 578.00000 560.31293 3.06% 17.4 420s 713063 8885 568.11389 37 59 578.00000 566.75240 1.95% 17.2 425s Explored 723627 nodes (12293156 simplex iterations) in 428.08 seconds Thread count was 4 (of 4 available processors)

Lopes Monash University Some Quadratic Assignment Formulations and their impact on Gurobi 23/25

slide-47
SLIDE 47

Agenda Problem Definition Branch and Bound Diagnosing and Improving our Formulation Conclusions Step 2: Smaller constants and more terms

A healthier tree

... Nodes | Current Node | Objective Bounds | Work Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time ... 41971 20620 578.17341 42 57 586.00000 426.84741 27.2% 26.7 60s 49310 23712 506.64911 34 68 586.00000 430.28747 26.6% 26.2 65s 56491 26513 580.49821 55 64 586.00000 433.86992 26.0% 25.9 70s ... 443823 85925 cutoff 35 578.00000 517.55057 10.5% 19.6 300s 454929 85247 572.52381 38 46 578.00000 518.97396 10.2% 19.5 305s 461098 84930 cutoff 29 578.00000 519.80133 10.1% 19.4 310s ... 676266 34174 cutoff 39 578.00000 552.77727 4.36% 17.6 410s 687701 27147 cutoff 50 578.00000 556.03894 3.80% 17.5 415s 699749 18846 cutoff 30 578.00000 560.31293 3.06% 17.4 420s 713063 8885 568.11389 37 59 578.00000 566.75240 1.95% 17.2 425s Explored 723627 nodes (12293156 simplex iterations) in 428.08 seconds Thread count was 4 (of 4 available processors)

Lopes Monash University Some Quadratic Assignment Formulations and their impact on Gurobi 23/25

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SLIDE 48

Agenda Problem Definition Branch and Bound Diagnosing and Improving our Formulation Conclusions

Modeling Tools and solvers are powerful – but not automatic yet

◮ I am not endorsing these formulations as practical ways for solving

QAPs

◮ Even stronger bounds computed using more sophisticated machinery are

needed for larger instances

◮ Modeling Languages allow experimentation with new formulations at

very low cost

◮ The general principle is to define constraints as tightly as possible.

◮ Dramatic improvements can be obtained with very little extra programming

effort.

◮ Your real world “branch” problem likely contains a “core” problem

(sequencing, coloring, QAP). Check the academic literature for strong formulations of those problems.

◮ Analytical results showing that one constraint is dominated by another are

common and appreciated in the academic literature.

◮ This formulation can still be improved. How? Lopes Monash University Some Quadratic Assignment Formulations and their impact on Gurobi 24/25

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SLIDE 49

Agenda Problem Definition Branch and Bound Diagnosing and Improving our Formulation Conclusions

Thanks for listening! Parting words: put some thought into the problem and experiment with different formulations. It produces results (and is fun to do!)

Lopes Monash University Some Quadratic Assignment Formulations and their impact on Gurobi 25/25